Finite Group Databasebeta

← #20280453

Subgroups — #20280453

Order 6784.

824 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133189C211yesmaximal, characteristic, complemented
2C2×#13133186C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
4#16810551C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810537C212yesmaximal, complemented
7#16810531C212yesmaximal, complemented
8#16810503C211yesmaximal, characteristic, complemented
9#16810500C211yesmaximal, characteristic, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C2×#120×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2211yescharacteristic, complemented
15C23×Dic53C2211yescharacteristic
16C23×Dic53C2211yescharacteristic
17#13133189—24no—
18#13133186—24no—
19C2×#11719919C2211yescharacteristic, complemented
20C2×#11719919C2212yescomplemented
21C2×C22⋊Dic53—24no—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C22⋊Dic53C2211yescharacteristic, complemented
27C2×C22⋊Dic53C2212yescomplemented
28C2×#11719921C2211yescharacteristic, complemented
29C2×#11719921C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32#13133124C2212yescomplemented
33C25×C53C2211yescharacteristic
34C22×D4×C53C2211yescharacteristic
35C22×D4×C53C2212yes—
36#120×C53—24no—
37C2×C22⋊C4×C53C2211yescharacteristic
38C2×C22⋊C4×C53C2212yes—
39C24×C53—22no—
40C24×C53—22no—
41C24×C53—42no—
42C24×C53—42no—
43C24×C53D414yescomplemented
44C24×C53C2311yescharacteristic
45C24×C53C2312yes—
46C24×C53C2311yescharacteristic
47C23×D53C2311yescharacteristic
48C2×D4×C53—24no—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2312yes—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—22no—
59C2×C22⋊D53—22no—
60C2×C22⋊D53—22no—
61C2×C22⋊D53—42no—
62C2×C22⋊D53—42no—
63C2×C22⋊D53—42no—
64C22×Dic53—22no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53D414yescomplemented
68C22×Dic53C2312yes—
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2312yes—
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22⋊C4×C53—24no—
74C22⋊C4×C53—22no—
75C22⋊Dic53—24no—
76C22⋊Dic53—22no—
77C22⋊Dic53—24no—
78C22⋊Dic53—24no—
79C22⋊Dic53—22no—
80C22⋊Dic53—42no—
81#11719921—22no—
82#11719921—42no—
83#11719919—22no—
84#11719919—42no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—28no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—48no—
100C23×C53—44no—
101C23×C53—44no—
102C23×C53—48no—
103C23×C53C2×D412yes—
104C23×C53C2×D414yes—
105C23×C53C2×D412yes—
106C23×C53#4112yes—
107C23×C53C2411yescharacteristic
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—22no—
112C22×D53—22no—
113D4×C53—48no—
114D4×C53—44no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C22⋊D53—44no—
122C22⋊D53—48no—
123C22⋊D53—44no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—21no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—44no—
136C2×Dic53—44no—
137C2×Dic53C2×D412yes—
138C2×Dic53C2×D412yes—
139C22×C53—22no—
140C22×C53—24no—
141C22×C53—22no—
142C22×C53—24no—
143C22×C53—24no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53—416no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—42no—
154C22×C53—41no—
155C22×C53—42no—
156C22×C53#12114yes—
157C22×C53C22×D411yescharacteristic
158C22×C53C22×D411yescharacteristic
159C22×C53C2×#4111yescharacteristic
160C22×C53H(5,𝔽2)12yes—
161C22×C53H(5,𝔽2)12yes—
162C2×D53—41no—
163C2×D53—42no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—42no—
167C4×C53—41no—
168C4×C53—42no—
169Dic53—22no—
170Dic53—22no—
171Dic53—41no—
172Dic53—42no—
173Dic53—42no—
174#2617—531nomaximal, Sylow
175C2×C53—24no—
176C2×C53—44no—
177C2×C53—81no—
178C2×C53C2×#12111yescharacteristic
179C2×C53#53412yes—
180C2×C53#53411yescharacteristic
181C2×C53#53511yescharacteristic
182C2×C53#53812yes—
183D53—81no—
184C2×#121—531no—
185C2×#120—531no—
186C2×#120D5311yescharacteristic, complemented
187C22×C22⋊C4—531no—
188#397—532no—
189#394—532no—
190#394—532no—
191#392—532no—
192#386—531no—
193#386—531no—
194#380—531no—
195C53#261711yescharacteristic, complemented, Sylow
196C25C2×D5311yescharacteristic
197C22×D4—531no—
198C22×D4—532no—
199C22×D4—531no—
200C22×D4C2×D5311yescharacteristic
201C22×D4C2×D5312yes—
202C23×C4—531no—
203C23×C4—531no—
204#121—1064no—
205#120—24no—
206#120—1064no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—532no—
216C2×C22⋊C4—532no—
217C2×C22⋊C4—1064no—
218C2×C22⋊C4C2×D5311yescharacteristic
219C2×C22⋊C4C2×D5312yes—
220#95—532no—
221#95—532no—
222#95—532no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24—531no—
228C24C22⋊D5314yescomplemented
229C24C22×D5311yescharacteristic
230C24C22×D5312yes—
231C24C22×D5311yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—1062no—
239C2×D4—1062no—
240C2×D4—1062no—
241C2×D4—1064no—
242C2×D4—1062no—
243C2×D4—2122no—
244C2×D4—2122no—
245C2×D4—2122no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—534no—
249C22×C4—532no—
250C22×C4—531no—
251C22×C4—532no—
252C22×C4—1062no—
253C22×C4—2122no—
254C22×C4—2122no—
255C22×C4C22×D5311yescharacteristic
256C22×C4C22×D5312yes—
257C4⋊C4—1062no—
258C4⋊C4—2122no—
259C22⋊C4—24no—
260C22⋊C4—22no—
261C22⋊C4—1062no—
262C22⋊C4—1064no—
263C22⋊C4—1064no—
264C22⋊C4—1062no—
265C22⋊C4—1064no—
266C22⋊C4—1062no—
267C22⋊C4—2122no—
268C22⋊C4—2122no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—1061no—
288C23—1061no—
289C23—1061no—
290C23—1062no—
291C23—1062no—
292C23D4×D5312yes—
293C23#1171994712yes—
294C23C2×C22⋊D5314yes—
295C23C2×C22⋊D5312yes—
296C23C23×D5311yescharacteristic
297D4—48no—
298D4—44no—
299D4—2124no—
300D4—2128no—
301D4—2124no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—532no—
309C2×C4—532no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1062no—
316C2×C4—1061no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—2124no—
322C2×C4—2124no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—24no—
328C22—24no—
329C22—24no—
330C22—416no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—2121no—
341C22—2122no—
342C22—2121no—
343C22—2121no—
344C22—2122no—
345C22#1313318914yes—
346C22C2×D4×D5311yescharacteristic
347C22C22×C22⋊D5311yescharacteristic
348C22C2×#1171994711yescharacteristic
349C22#1313326212yes—
350C22#1313326212yes—
351C4—41no—
352C4—42no—
353C4—1062no—
354C4—1062no—
355C4—2121no—
356C4—2122no—
357C4—2122no—
358C2—24no—
359C2—44no—
360C2—81no—
361C2—4241no—
362C2C2×#1313318911yescharacteristic
363C2#1681118212yes—
364C2#1681118311yescharacteristic
365C2#1681118811yescharacteristic
366C2#1681120812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2617
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133186C2semidirect
3. C22×C22⋊Dic53C2semidirect
4. #16810551C2semidirect
5. #16810538C2semidirect
6. #16810537C2semidirect
7. #16810531C2semidirect
8. #16810503C2semidirect
9. #16810500C2semidirect
10. #16810481C2semidirect
11. C2×#120×C53C2semidirect
14. C22×C22⋊D53C22semidirect
19. C2×#11719919C22semidirect
20. C2×#11719919C22semidirect
26. C2×C22⋊Dic53C22semidirect
27. C2×C22⋊Dic53C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
43. C24×C53D4semidirect
67. C22×Dic53D4semidirect
186. C2×#120D53semidirect
195. C53#2617semidirect
228. C24C22⋊D53semidirect